In this section we show that the method of linear variations (also called
the Ritz method [12]) is equivalent to the matrix formulation 
of the Schrödinger equation 
. Our treatment is similar to that of Szabo and Ostlund
[1], p. 116.
The linear variation method states that, given the linear expansion
we vary the coefficients 
so that we minimize 
. We begin by requiring that the
wavefunction be normalized so that 
.
Normalization means that we cannot minimize E simply by solving
because the 's are not independent.
In this case we have a constrained
minimization, so we apply Lagrange's method of undetermined multipliers,
and we minimize the functional
which has the same minimum as E when 
is normalized.
When we substitute equation (3.1) into equation (3.3),
we obtain
which we may rewrite as
where of course 
and
. Now set the first variation in
equal to zero:
Since the summations run over all i and j, and since 
and 
, we can simplify to
Since each term is a sum of a number and its complex conjugate, the
imaginary parts will cancel. However, the real part will not necessarily
be zero; in fact, since all the 
's are arbitrary (that is the
whole point of using Lagrange's method), then for 
to be
zero, the term in brackets must be zero. We may rewrite this condition as
a matrix equation
If the basis functions { 
} are chosen orthonormal (as is
usually the case), then 
the identity matrix, and we
have . Of course 
is the column-vector
representation of in the basis { }. We thus
have two equivalent ways of viewing a CI--either as the matrix formulation
of the Schrödinger equation within the given linear vector space of
N-electron basis functions, or as the minimization of the energy
with respect to the linear expansion coefficients of
(3.1), subject to the constraint that the wavefunction remain
normalized. Another way of viewing the results of this section is to note
that only eigenvectors of the Hamiltonian matrix 
are stable with
respect to variations in the linear expansion coefficients.
At this point it is reasonable to ask why we wish to minimize the
energy by varying the coefficients in equation (3.1).
How do we know that this will give us the best estimate of the wavefunction?
There are two answers to this. First, as we have just shown, minimizing the
energy by variation of the linear expansion coefficients gives the
Schrödinger equation in matrix form; thus the procedure is justified
a posteriori by the validity of its result. The other reason is that,
for the ground state, the linear expansion in equation (3.1)
gives an expectation value for the energy E which is always an
upper bound to the exact nonrelativistic ground state energy
. We will prove this assertion in the next section; the
result is called the Variational Theorem.
The best estimate of E, then, is the minimum value which
can be obtained by varying the coefficients in equation (3.1)
(while also maintaining normalization). These arguments also hold for
excited states, so long as each excited state is made orthogonal to
all lower states.